# American Institute of Mathematical Sciences

January  2020, 16(1): 371-386. doi: 10.3934/jimo.2018157

## Analysis of strategic customer behavior in fuzzy queueing systems

 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author.Email address: math_zjc@csu.edu.cn (Jingchuan Zhang)

Received  June 2018 Published  January 2020 Early access  September 2018

Fund Project: This work is partially supported by the National Natural Science Foundation of China (11671404), and the Fundamental Research Funds for the Central Universities of Central South University (2017zzts061, 2017zzts386).

This paper analyzes the optimal and equilibrium strategies in fuzzy Markovian queues where the system parameters are all fuzzy numbers. In this work, tools from both fuzzy logic and queuing theory have been used to investigate the membership functions of the optimal and equilibrium strategies in both observable and unobservable cases. By Zadeh's extension principle and $α$-cut approach, we formulate a pair of parametric nonlinear programs to describe the family of crisp strategy. Then the membership functions of the strategies in single and multi-server models are derived. Furthermore, the grated mean integration method is applied to find estimate of the equilibrium strategy in the fuzzy sense. Finally, numerical examples are solved successfully to illustrate the validity of the proposed approach and a sensitivity analysis is performed, which show the relationship of these strategies and social benefits. Our finding reveals that the value of equilibrium and optimal strategies have no deterministic relationship, which are different from the results in the corresponding crisp queues. Since the performance measures of such queues are expressed by fuzzy numbers rather than by crisp values, the system managers could get more precise information.

Citation: Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial and Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157
##### References:
 [1] O. Bountali and A. Economou, Equilibrium joining strategies in batch service queueing systems, European Journal of Operational Research, 260 (2017), 1142-1151.  doi: 10.1016/j.ejor.2017.01.024. [2] S.-P. Chen, Time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness, European Journal of Operational Research, 255 (2016), 834-844.  doi: 10.1016/j.ejor.2016.06.021. [3] D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, vol. 144, Academic Press, 1980. [4] A. Economou and A. Manou, Strategic behavior in an observable fluid queue with an alternating service process, European Journal of Operational Research, 254 (2016), 148-160.  doi: 10.1016/j.ejor.2016.03.046. [5] P. Guo and R. Hassin, Strategic behavior and social optimization in markovian vacation queues: The case of heterogeneous customers, European Journal of Operational Research, 222 (2012), 278-286.  doi: 10.1016/j.ejor.2012.05.026. [6] R. Hassin and M. Haviv, To queue or not to queue: Equilibrium behavior in queueing systems, vol. 59, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0. [7] F. Jolai, S. M. Asadzadeh, R. Ghodsi and S. Bagheri-Marani, A multi-objective fuzzy queuing priority assignment model, Applied Mathematical Modelling, 40 (2016), 9500-9513.  doi: 10.1016/j.apm.2016.06.024. [8] J.-C. Ke, H.-I. Huang and C.-H. Lin, Parametric programming approach for batch arrival queues with vacation policies and fuzzy parameters, Applied Mathematics and Computation, 180 (2006), 217-232.  doi: 10.1016/j.amc.2005.11.144. [9] J.-C. Ke, H.-I. Huang and C.-H. Lin, On retrial queueing model with fuzzy parameters, Physica A: Statistical Mechanics and its Applications, 374 (2007), 272-280.  doi: 10.1016/j.physa.2006.05.057. [10] J.-C. Ke, H.-I. Huang and C.-H. Lin, Analysis on a queue system with heterogeneous servers and uncertain patterns, Journal of Industrial & Management Optimization, 6 (2010), 57-71.  doi: 10.3934/jimo.2010.6.57. [11] J.-C. Ke and C.-H. Lin, Fuzzy analysis of queueing systems with an unreliable server: A nonlinear programming approach, Applied Mathematics and Computation, 175 (2006), 330-346.  doi: 10.1016/j.amc.2005.07.024. [12] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, vol. 4, Prentice Hall New Jersey, 1995. [13] R.-J. Li and E. Lee, Analysis of fuzzy queues, Computers & Mathematics with Applications, 17 (1989), 1143-1147.  doi: 10.1016/0898-1221(89)90044-8. [14] Y. Ma, Z. Liu and Z. G. Zhang, Equilibrium in vacation queueing system with complementary services, Quality Technology & Quantitative Management, 14 (2017), 114-127.  doi: 10.1080/16843703.2016.1191172. [15] G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables, Computers & Industrial Engineering, 64 (2013), 190-199.  doi: 10.1016/j.cie.2012.09.003. [16] P. Naor, The regulation of queue size by levying tolls, Econometrica: Journal of the Econometric Society, 37 (1969), 15-24.  doi: 10.2307/1909200. [17] L. J. Ratliff, C. Dowling, E. Mazumdar and B. Zhang, To observe or not to observe: Queuing game framework for urban parking, in Decision and Control (CDC), 2016 IEEE 55th Conference on, IEEE, 2016, 5286-5291. doi: 10.1109/CDC.2016.7799079. [18] M. Saffari, S. Asmussen and R. Haji, The M/M/1 queue with inventory, lost sale, and general lead times, Queueing Systems, 75 (2013), 65-77.  doi: 10.1007/s11134-012-9337-3. [19] R. Shone, V. A. Knight and J. E. Williams, Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior, European Journal of Operational Research, 227 (2013), 133-141.  doi: 10.1016/j.ejor.2012.12.016. [20] E. Simhon, Y. Hayel, D. Starobinski and Q. Zhu, Optimal information disclosure policies in strategic queueing games, Operations Research Letters, 44 (2016), 109-113.  doi: 10.1016/j.orl.2015.12.005. [21] S. Stidham Jr, Optimal Design of Queueing Systems, CRC Press, 2009. doi: 10.1201/9781420010008. [22] B. Vahdani, R. Tavakkoli-Moghaddam and F. Jolai, Reliable design of a logistics network under uncertainty: A fuzzy possibilistic-queuing model, Applied Mathematical Modelling, 37 (2013), 3254-3268.  doi: 10.1016/j.apm.2012.07.021. [23] J. Wang and F. Zhang, Strategic joining in M/M/1 retrial queues, European Journal of Operational Research, 230 (2013), 76-87.  doi: 10.1016/j.ejor.2013.03.030. [24] Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold, Journal of Industrial and Management Optimization, 5 (2009), 467-479.  doi: 10.3934/jimo.2009.5.467. [25] F. Zhang, J. Wang and B. Liu, Equilibrium joining probabilities in observable queues with general service and setup times, Journal of Industrial and Management Optimization, 9 (2013), 901-917.  doi: 10.3934/jimo.2013.9.901. [26] R. Zhang, Y. A. Phillis and V. S. Kouikoglou, Fuzzy Control of Queuing Systems, Springer Science & Business Media, 2005. [27] S. Zhu and J. Wang, Strategic behavior and optimal strategies in an M/G/1 queue with bernoulli vacations, Journal of Industrial & Management Optimization, (2008), 56-64.  doi: 10.3934/jimo.2018008. [28] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Second edition, Kluwer Academic Publishers, Boston, MA, 1992.

show all references

##### References:
 [1] O. Bountali and A. Economou, Equilibrium joining strategies in batch service queueing systems, European Journal of Operational Research, 260 (2017), 1142-1151.  doi: 10.1016/j.ejor.2017.01.024. [2] S.-P. Chen, Time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness, European Journal of Operational Research, 255 (2016), 834-844.  doi: 10.1016/j.ejor.2016.06.021. [3] D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, vol. 144, Academic Press, 1980. [4] A. Economou and A. Manou, Strategic behavior in an observable fluid queue with an alternating service process, European Journal of Operational Research, 254 (2016), 148-160.  doi: 10.1016/j.ejor.2016.03.046. [5] P. Guo and R. Hassin, Strategic behavior and social optimization in markovian vacation queues: The case of heterogeneous customers, European Journal of Operational Research, 222 (2012), 278-286.  doi: 10.1016/j.ejor.2012.05.026. [6] R. Hassin and M. Haviv, To queue or not to queue: Equilibrium behavior in queueing systems, vol. 59, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0. [7] F. Jolai, S. M. Asadzadeh, R. Ghodsi and S. Bagheri-Marani, A multi-objective fuzzy queuing priority assignment model, Applied Mathematical Modelling, 40 (2016), 9500-9513.  doi: 10.1016/j.apm.2016.06.024. [8] J.-C. Ke, H.-I. Huang and C.-H. Lin, Parametric programming approach for batch arrival queues with vacation policies and fuzzy parameters, Applied Mathematics and Computation, 180 (2006), 217-232.  doi: 10.1016/j.amc.2005.11.144. [9] J.-C. Ke, H.-I. Huang and C.-H. Lin, On retrial queueing model with fuzzy parameters, Physica A: Statistical Mechanics and its Applications, 374 (2007), 272-280.  doi: 10.1016/j.physa.2006.05.057. [10] J.-C. Ke, H.-I. Huang and C.-H. Lin, Analysis on a queue system with heterogeneous servers and uncertain patterns, Journal of Industrial & Management Optimization, 6 (2010), 57-71.  doi: 10.3934/jimo.2010.6.57. [11] J.-C. Ke and C.-H. Lin, Fuzzy analysis of queueing systems with an unreliable server: A nonlinear programming approach, Applied Mathematics and Computation, 175 (2006), 330-346.  doi: 10.1016/j.amc.2005.07.024. [12] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, vol. 4, Prentice Hall New Jersey, 1995. [13] R.-J. Li and E. Lee, Analysis of fuzzy queues, Computers & Mathematics with Applications, 17 (1989), 1143-1147.  doi: 10.1016/0898-1221(89)90044-8. [14] Y. Ma, Z. Liu and Z. G. Zhang, Equilibrium in vacation queueing system with complementary services, Quality Technology & Quantitative Management, 14 (2017), 114-127.  doi: 10.1080/16843703.2016.1191172. [15] G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables, Computers & Industrial Engineering, 64 (2013), 190-199.  doi: 10.1016/j.cie.2012.09.003. [16] P. Naor, The regulation of queue size by levying tolls, Econometrica: Journal of the Econometric Society, 37 (1969), 15-24.  doi: 10.2307/1909200. [17] L. J. Ratliff, C. Dowling, E. Mazumdar and B. Zhang, To observe or not to observe: Queuing game framework for urban parking, in Decision and Control (CDC), 2016 IEEE 55th Conference on, IEEE, 2016, 5286-5291. doi: 10.1109/CDC.2016.7799079. [18] M. Saffari, S. Asmussen and R. Haji, The M/M/1 queue with inventory, lost sale, and general lead times, Queueing Systems, 75 (2013), 65-77.  doi: 10.1007/s11134-012-9337-3. [19] R. Shone, V. A. Knight and J. E. Williams, Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior, European Journal of Operational Research, 227 (2013), 133-141.  doi: 10.1016/j.ejor.2012.12.016. [20] E. Simhon, Y. Hayel, D. Starobinski and Q. Zhu, Optimal information disclosure policies in strategic queueing games, Operations Research Letters, 44 (2016), 109-113.  doi: 10.1016/j.orl.2015.12.005. [21] S. Stidham Jr, Optimal Design of Queueing Systems, CRC Press, 2009. doi: 10.1201/9781420010008. [22] B. Vahdani, R. Tavakkoli-Moghaddam and F. Jolai, Reliable design of a logistics network under uncertainty: A fuzzy possibilistic-queuing model, Applied Mathematical Modelling, 37 (2013), 3254-3268.  doi: 10.1016/j.apm.2012.07.021. [23] J. Wang and F. Zhang, Strategic joining in M/M/1 retrial queues, European Journal of Operational Research, 230 (2013), 76-87.  doi: 10.1016/j.ejor.2013.03.030. [24] Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold, Journal of Industrial and Management Optimization, 5 (2009), 467-479.  doi: 10.3934/jimo.2009.5.467. [25] F. Zhang, J. Wang and B. Liu, Equilibrium joining probabilities in observable queues with general service and setup times, Journal of Industrial and Management Optimization, 9 (2013), 901-917.  doi: 10.3934/jimo.2013.9.901. [26] R. Zhang, Y. A. Phillis and V. S. Kouikoglou, Fuzzy Control of Queuing Systems, Springer Science & Business Media, 2005. [27] S. Zhu and J. Wang, Strategic behavior and optimal strategies in an M/G/1 queue with bernoulli vacations, Journal of Industrial & Management Optimization, (2008), 56-64.  doi: 10.3934/jimo.2018008. [28] H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Second edition, Kluwer Academic Publishers, Boston, MA, 1992.
The approximate membership function of optimal and equilibrium threshold $n$.
The approximate membership function of optimal and equilibrium arrival rate $\lambda$.
The approximate membership function of optimal social benefit $S$.
Fuzzy trapezoidal value for the input parameters $\tilde{\Lambda}$
 $\tilde{\Lambda}$ $P(\tilde{\Lambda})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (1.0, 2.0, 3.0, 4.0) $2.50$ $22$ $14$ $2.50$ $2.50$ (1.1, 2.2, 3.3, 4.4) $2.75$ $22$ $13$ $2.75$ $2.75$ (1.2, 2.4, 3.6, 4.8) $3.00$ $22$ $13$ $3.00$ $3.00$ (1.3, 2.6, 3.9, 5.2) $3.25$ $22$ $12$ $3.25$ $3.25$ (1.4, 2.8, 4.2, 5.6) $3.50$ $22$ $11$ $3.50$ $3.50$ (1.5, 3.0, 4.5, 6.0) $3.75$ $22$ $10$ $3.75$ $3.75$
 $\tilde{\Lambda}$ $P(\tilde{\Lambda})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (1.0, 2.0, 3.0, 4.0) $2.50$ $22$ $14$ $2.50$ $2.50$ (1.1, 2.2, 3.3, 4.4) $2.75$ $22$ $13$ $2.75$ $2.75$ (1.2, 2.4, 3.6, 4.8) $3.00$ $22$ $13$ $3.00$ $3.00$ (1.3, 2.6, 3.9, 5.2) $3.25$ $22$ $12$ $3.25$ $3.25$ (1.4, 2.8, 4.2, 5.6) $3.50$ $22$ $11$ $3.50$ $3.50$ (1.5, 3.0, 4.5, 6.0) $3.75$ $22$ $10$ $3.75$ $3.75$
Fuzzy trapezoidal value for the input parameters $\tilde{\mu}$
 $\tilde{\mu}$ $P(\tilde{\mu})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (5.0, 6.0, 7.0, 8.0) $6.50$ $22.75$ $14$ $2.5$ $2.5$ (5.5, 6.6, 7.7, 8.8) $7.15$ $25.03$ $16$ $2.5$ $2.5$ (6.0, 7.2, 8.4, 9.6) $7.80$ $27.30$ $19$ $2.5$ $2.5$ (6.5, 7.8, 9.1, 10.4) $8.45$ $29.58$ $21$ $2.5$ $2.5$ (7.0, 8.4, 9.8, 11.2) $9.10$ $31.85$ $23$ $2.5$ $2.5$ (7.5, 9.0, 10.5, 12.0) $9.75$ $34.13$ $25$ $2.5$ $2.5$
 $\tilde{\mu}$ $P(\tilde{\mu})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (5.0, 6.0, 7.0, 8.0) $6.50$ $22.75$ $14$ $2.5$ $2.5$ (5.5, 6.6, 7.7, 8.8) $7.15$ $25.03$ $16$ $2.5$ $2.5$ (6.0, 7.2, 8.4, 9.6) $7.80$ $27.30$ $19$ $2.5$ $2.5$ (6.5, 7.8, 9.1, 10.4) $8.45$ $29.58$ $21$ $2.5$ $2.5$ (7.0, 8.4, 9.8, 11.2) $9.10$ $31.85$ $23$ $2.5$ $2.5$ (7.5, 9.0, 10.5, 12.0) $9.75$ $34.13$ $25$ $2.5$ $2.5$
Fuzzy trapezoidal value for the input parameters $\tilde{R}$
 $\tilde{R}$ $P(\tilde{R})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (10.0, 15.0, 20.0, 25.0) $17.50$ $22$ $14$ $2.5$ $2.5$ (11.0, 16.5, 22.0, 27.5) $19.25$ $25$ $16$ $2.5$ $2.5$ (12.0, 18.0, 24.0, 30.0) $21.00$ $27$ $17$ $2.5$ $2.5$ (13.0, 19.5, 26.0, 32.5) $22.75$ $29$ $18$ $2.5$ $2.5$ (14.0, 21.0, 28.0, 35.0) $24.50$ $31$ $20$ $2.5$ $2.5$ (15.0, 22.5, 30.0, 37.5) $26.25$ $34$ $21$ $2.5$ $2.5$
 $\tilde{R}$ $P(\tilde{R})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (10.0, 15.0, 20.0, 25.0) $17.50$ $22$ $14$ $2.5$ $2.5$ (11.0, 16.5, 22.0, 27.5) $19.25$ $25$ $16$ $2.5$ $2.5$ (12.0, 18.0, 24.0, 30.0) $21.00$ $27$ $17$ $2.5$ $2.5$ (13.0, 19.5, 26.0, 32.5) $22.75$ $29$ $18$ $2.5$ $2.5$ (14.0, 21.0, 28.0, 35.0) $24.50$ $31$ $20$ $2.5$ $2.5$ (15.0, 22.5, 30.0, 37.5) $26.25$ $34$ $21$ $2.5$ $2.5$
Fuzzy trapezoidal value for the input parameters $\tilde{C}$
 $\tilde{C}$ $P(\tilde{C})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (2.0, 4.0, 6.0, 8.0) $5.00$ $22$ $14$ $2.5$ $2.5$ (2.2, 4.4, 6.6, 8.8) $5.50$ $20$ $13$ $2.5$ $2.5$ (2.4, 4.8, 7.2, 9.6) $6.00$ $18$ $12$ $2.5$ $2.5$ (2.6, 5.2, 7.8, 10.4) $6.50$ $17$ $11$ $2.5$ $2.5$ (2.8, 5.6, 8.4, 11.2) $7.00$ $16$ $10$ $2.5$ $2.5$ (3.0, 6.0, 7.0, 12.0) $6.83$ $16$ $10$ $2.5$ $2.5$
 $\tilde{C}$ $P(\tilde{C})$ $n_{e}$ $n_{*}$ $\lambda_{e}$ $\lambda_{*}$ (2.0, 4.0, 6.0, 8.0) $5.00$ $22$ $14$ $2.5$ $2.5$ (2.2, 4.4, 6.6, 8.8) $5.50$ $20$ $13$ $2.5$ $2.5$ (2.4, 4.8, 7.2, 9.6) $6.00$ $18$ $12$ $2.5$ $2.5$ (2.6, 5.2, 7.8, 10.4) $6.50$ $17$ $11$ $2.5$ $2.5$ (2.8, 5.6, 8.4, 11.2) $7.00$ $16$ $10$ $2.5$ $2.5$ (3.0, 6.0, 7.0, 12.0) $6.83$ $16$ $10$ $2.5$ $2.5$
 [1] Wai-Ki Ching, Tang Li, Sin-Man Choi, Issic K. C. Leung. A tandem queueing system with applications to pricing strategy. Journal of Industrial and Management Optimization, 2009, 5 (1) : 103-114. doi: 10.3934/jimo.2009.5.103 [2] Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial and Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581 [3] Yahia Zare Mehrjerdi. A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 513-535. doi: 10.3934/naco.2021019 [4] Jinyan Wang, Yanni Xiao, Robert A. Cheke. Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5849-5870. doi: 10.3934/dcdsb.2019109 [5] Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036 [6] Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial and Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73 [7] Wenjuan Jia, Yingjie Deng, Chenyang Xin, Xiaodong Liu, Witold Pedrycz. A classification algorithm with Linear Discriminant Analysis and Axiomatic Fuzzy Sets. Mathematical Foundations of Computing, 2019, 2 (1) : 73-81. doi: 10.3934/mfc.2019006 [8] Qi Wang, Yue Zhou. Sets of zero-difference balanced functions and their applications. Advances in Mathematics of Communications, 2014, 8 (1) : 83-101. doi: 10.3934/amc.2014.8.83 [9] Kanat Abdukhalikov, Duy Ho. Vandermonde sets, hyperovals and Niho bent functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021048 [10] Xue-Yan Wu, Zhi-Ping Fan, Bing-Bing Cao. Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1999-2027. doi: 10.3934/jimo.2019040 [11] Haijiao Li, Kuan Yang, Guoqing Zhang. Optimal pricing strategy in a dual-channel supply chain: A two-period game analysis. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022072 [12] Yu Li, Kok Lay Teo, Shuhua Zhang. A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022105 [13] Sin-Man Choi, Ximin Huang, Wai-Ki Ching. Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment. Journal of Industrial and Management Optimization, 2012, 8 (2) : 299-323. doi: 10.3934/jimo.2012.8.299 [14] Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial and Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133 [15] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [16] Henk van Tilborg, Josef Pieprzyk, Ron Steinfeld, Huaxiong Wang. New constructions of anonymous membership broadcasting schemes. Advances in Mathematics of Communications, 2007, 1 (1) : 29-44. doi: 10.3934/amc.2007.1.29 [17] Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041 [18] Xuemei Li, Rafael de la Llave. Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 623-641. doi: 10.3934/dcdss.2010.3.623 [19] Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353 [20] Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217

2021 Impact Factor: 1.411